UPSC NDA I 2025 Mathematics Paper

We are given the equation for the ratio of sides using the Sine Rule:
‌

a

sin‌(3x)

=‌

b

sin‌(x)

‌

a

b

=‌

sin‌(3x)

sin‌(x)

Step 3: Use the identity for sin‌(3x), which is sin‌(3x)=3sin‌(x)−4sin‌3(x), and substitute it into the equation:
‌2=‌

3sin‌(x)−4sin‌3(x)

sin‌(x)

‌2sin‌(x)=3sin‌(x)−4sin‌3(x)
‌−sin‌(x)+4sin‌3(x)=0
‌sin‌(x)(4sin‌2(x)−1)=0
We have two possible solutions for this equation:
sin‌(x)=0, which gives x=0∘ (not valid in this case).
4sin‌2(x)−1=0, which simplifies to:
‌sin‌2(x)=‌

1

4

‌‌⇒‌‌sin‌(x)=‌

1

2

‌x=30∘